56 Prof. H. Lamb on the Theory of Electric 



the thickness of the stratum in question. Since these equa- 

 tions are only true in a statistical sense, when the linear 

 elements dx, dy, dz are taken to be large in comparison with 

 the average distance between neighbouring molecules, whereas 

 the thickness of the stratum is almost certainly not more than 

 a very moderate multiple of this distance, it seems doubtful 

 whether they can fairly be pressed into service in the manner 

 indicated. 



Although we have only somewhat vague probabilities to 

 guide us, it appears reasonable to suppose, from what we 

 know of contact differences of potential in cases where they 

 can be measured, that the ratio E/D will not very greatly 

 exceed or fall below unity ; that it will lie, say, between about 

 •1 and 10. If this be so, the comparison of our theory with 

 the observations entitles us to say that the sliding coefficient 

 I is at all events of the same order of magnitude as d. If 

 for water in contact with glass I were equal to 10 -8 centim., 

 this would make 



(8=/a//=1-4x10 6 C.G.S.; 



in other words, the shearing-stress necessary (in the absence 

 of electrical surface-forces) to produce a sliding of one centi- 

 metre per second would be 1*4 megadynes per square centim. 

 It follows that the effects of slipping would be utterly insen- 

 sible in ordinary hydrodynamical questions, e. g. the experi- 

 ments of Poiseuille. The slipping leads to appreciable results 

 in the cases at present in view, only in consequence of the 

 relatively enormous electrical forces acting on the superficial 

 film, and dragging the fluid (as it were) by the skin, through 

 the tube. 



The formulas (6) may be written: — 



Flux of liquid _ otE 

 Flux of electricity /3 ' 



In this form it can be shown to be true, under a certain 

 restriction, for a tube of varying section, for a network of 

 tubes, and even for the labyrinth of channels contained in the 

 walls of a porous vessel, provided no difference of pressure be 

 allowed to establish itself on the two sides. 



Let cf) denote, as before, the electric potential at any point 

 of the fluid. It will appear that all the conditions of our 

 problem will be satisfied if we suppose the motion of the fluid 

 to be irrotational, the velocity-potential % being everywhere 

 proportional to <£. 



Since V 2 % = ; the equations of steady small motion of a 

 viscous liquid, viz. 



